Theory If arg(w)=π4\arg(w)=\frac{\pi}{4}arg(w)=4π, then tanarg(w)=1\tan\arg(w)=1tanarg(w)=1 and (when Re(w),Im(w)\operatorname{Re}(w),\operatorname{Im}(w)Re(w),Im(w) have the same sign): Im(w)=Re(w)\operatorname{Im}(w)=\operatorname{Re}(w)Im(w)=Re(w) To simplify a complex fraction a+bic+di\dfrac{a+bi}{c+di}c+dia+bi, multiply numerator and denominator by c−dic-dic−di.